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The T model microstructure shown in Figure 11 is very similar to the I model except that the lower section of the image has been changed from C-S-H to pore fluid. The crossbar in this case has extended from the cement grain into a pore, but not far enough to isolate one side of the pore from the other. This leaves the high conductivity pore phase continuous, as is the case in real 3-D cement paste. The impedance response for this system is shown in Figure 12 for six different crossbar thicknesses. As with the I model, the composite resistance increases with increasing crossbar thickness; however, the magnitude of the resistance is much less. The size of the low frequency arc is also much smaller than in the I model, and the high frequency arc is much more visible. The value of k_{eff} associated with the low frequency arc follows the same trend as in the I model (see Figure 8), but its value is higher, especially at smaller thickness values.
Figure 11: Schematic of T model, shown with a crossbar thickness, t, of two pixels.
Figure 13a shows a picture of the T model, with the blue phase being pore fluid and the red phase being C-S-H. The thickness (t) is 1 pixel. Figure 13b shows a map of DC current intensity for the T model. It is apparent that a significant amount of current simply flows around the crossbar through the much more conductive portion of the composite, thus significantly reducing the composite resistance, and, therefore, the size of the low frequency arc. Distribution functions of effective D.C. conductivities for the T model with a crossbar thickness of 1 are shown in Figure 14 for C-S-H (a) and pore fluid (b). Table 3 lists the average single phase effective conductivities for the T model at different thicknesses, analogous to Table 2. It is evident from Figure 14 (b) and Figure 13 that the pore fluid in the T model has a much wider distribution of effective conductivities than the pore fluid in the I model. The distribution of effective conductivities for C-S-H in the T model is not much different than in the I model.
t(pixels) | Effective conductivity (C-S-H) | Effective conductivity (pore fluid) | V_{f} (C-S-H) | V_{f} (pore fluid) | Composite conductivity | R |
1 | 0.01984 | 0.62668 | 0.275 | 0.725 | 0.460 | 2.175 |
2 | 0.01948 | 0.59377 | 0.300 | 0.700 | 0.421 | 2.373 |
3 | 0.01876 | 0.57840 | 0.325 | 0.675 | 0.397 | 2.522 |
4 | 0.01800 | 0.56933 | 0.350 | 0.650 | 0.376 | 2.657 |
5 | 0.01725 | 0.56479 | 0.375 | 0.625 | 0.359 | 2.782 |
6 | 0.01653 | 0.56300 | 0.400 | 0.600 | 0.344 | 2.903 |
Table 3: Effective conductivity data for the T model. σ_{eff} is the effective conductivity for a given phase, as defined in eqs. (3)-(5), V_{f} is the volume fraction of a phase, σ_{comp} is the composite DC conductivity, and R = 1/σ_{comp} = the bulk resistance. |
Figure 13a: T model with a crossbar thickness of 1. Blue=pore fluid, red=C-S-H.
Figure 13b: Corresponding current intensity plot for system of Fig. 13a, where white represents the highest value of current.
Figure 14b: Effective conductivity distribution function (pore fluid phase) in the T model (crossbar thickness of 1).
The flow of current around the crossbar in Figure 13 is not too surprising, and raises the question of why, since the crossbar is providing the mechanism for dielectric amplification, should the T model have a higher value of k_{eff} than the I model?
The answer to this question lies in the effective conductivities of the crossbar pixels for each model. The following explanation, applicable to any crossbar thickness, compares the effective conductivities of the crossbar pixels for both the I and the T models, for a thickness of 1. Consider the crossbar pixels to be numbered 1 through 10, starting with the first pixel of the crossbar nearest the top, as pictured in Figures 6 and 11.
Figure 15 shows a plot of effective conductivity versus the position of the crossbar pixel for both the I and T models when the applied frequency is zero (DC). As expected, the effective conductivity of crossbar pixels in the I model is higher, and relatively constant along the bar. The crossbar pixels of the T model, on the other hand, have overall lower effective conductivities, and the pixels toward the end of the crossbar have significantly lower effective conductivities than those near the top. Near the top of the crossbar it is easier for the current to go through the crossbar than travel all the way to the bottom to go around the crossbar, which will tend to increase the effective conductivity near the top. Near the bottom of the crossbar, however, current may easily travel around the tip through the more conductive pore fluid, thus reducing the effective conductivity of the pixels near the bottom of the crossbar.
The situation is quite different for the two models at the peak frequency ω_{o}. Figure 16 shows the effective conductivity (both real and imaginary) for the crossbar pixels at the peak of the low frequency arc for each model. In this case, the effective conductivities of the crossbar pixels in the T model are higher than those of the I model for both real and imaginary components. The geometry of the T model is such that the microstructural feature responsible for dielectric amplification (crossbar) has a higher effective conductivity at the peak frequency than its counterpart in the I model. In other words, more current passes through the crossbar in the T model, producing greater dielectric amplification. It is important to recall that the T model has a well-percolated high DC conductivity phase. In fact, at least for the I and T models, a percolated high DC conductivity phase actually seems produce higher values of k_{eff}.
Figure 16: Effective conductivity as a function of pixel position along the crossbar at the peak frequency (I - 200 kHz, T - 1 MHz) with a crossbar thickness of 1 pixel.